Question

Trigonometric Ratios Express the lengths a,b,c, and d in the figure in terms of the trigonometric ratios of$\theta$.

Short answer

The following expressions are obtained from the trigonometric ratios of in the given triangle:

$\begin{array}{l}\begin{array}{c}\mathbf{a}\mathbf{=}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\theta }\\ \mathbf{b}\mathbf{=}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{\theta }\end{array}\\ \begin{array}{l}\mathbf{c}\mathbf{=}\mathbf{s}\mathbf{e}\mathbf{c}\mathbf{\theta }\\ \mathbf{d}\mathbf{=}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\theta }\end{array}\end{array}$

Step-by-step solution

Step 1. Given information.

The dimensions of a,b,c,d and $\mathit{\theta }$ in the given diagram.

Step 2. Applying the known trigonometric ratios to the right-angled triangles in the given picture to deduce the relation between a,b,c,d and θ.  

In the inner triangle OGP,

we find that, $\cos \theta =\frac{PG}{OG}$since

OG is the radius, OG=1

$\begin{array}{l}\Rightarrow \cos \theta =d\\ Similarly\\ \sin \theta =\frac{OP}{OG}\\ \Rightarrow \sin \theta =a\\ Butintriangle\\ GMN,\\ GN=1\\ sinceradius,thus,\end{array}$

$\begin{array}{l}\cos \theta =\frac{GN}{GM}\\ \Rightarrow \cos \theta =\frac{1}{c}\\ \Rightarrow c=\sec \theta \end{array}$

Step3. Finding the value of b

Similarly,

$\begin{array}{c}\sin \theta =\frac{MN}{GM}\\ \Rightarrow \sin \theta =\frac{b}{c}\\ \Rightarrow \sin \theta =b\cos \theta \\ \Rightarrow b=\frac{\sin \theta }{\cos \theta }\\ \Rightarrow b=\tan \theta \end{array}$